(a) Simplify : (frac{1frac{1}{4} + frac{7}{9}}{1frac{4}{9} – 2frac{2}{3} times frac{9}{64}})
(b) Given that (sin x = frac{2}{3}), evaluate, leaving your answer in surd form and without using tables or calculator, (tan x – cos x).
Explanation
(a) (frac{1frac{1}{4} + frac{7}{9}}{1frac{4}{9} – 2frac{2}{3} times frac{9}{64}})
(1frac{1}{4} + frac{7}{9} = frac{5}{4} + frac{7}{9})
= (frac{45 + 28}{36})
= (frac{73}{36})
(1frac{4}{9} – 2frac{2}{3} times frac{9}{64} = frac{13}{9} – (frac{8}{3} times frac{9}{64}))
= (frac{13}{9} – frac{3}{8})
= (frac{104 – 27}{72})
= (frac{77}{72})
(therefore frac{1frac{1}{4} + frac{7}{9}}{1frac{4}{9} – 2frac{2}{3} times frac{9}{64}} = frac{73}{36} div frac{77}{72})
= (frac{73}{36} times frac{72}{77})
= (frac{146}{77})
= (1frac{69}{77})
(b) (sin x = frac{2}{3})
In (Delta ABC, 3^{2} = 2^{2} + |BC|^{2})
(therefore |BC|^{2} = 9 – 4 = 5)
(|BC| = sqrt{5})
(tan x = frac{2}{sqrt{5}} ; cos x = frac{sqrt{5}}{3})
(therefore tan x – cos x = frac{2}{sqrt{5}} – frac{sqrt{5}}{3})
= (frac{6 – 5}{3sqrt{5}})
= (frac{1}{3sqrt{5}})
Rationalizing, we have
(frac{1}{3sqrt{5}} times frac{sqrt{5}}{sqrt{5}} = frac{sqrt{5}}{15})